Jonathan. Frech’s WebBlog

Prime-Generating Formula (#114)

Jonathan Frech

(April Fools’!) I came up with this interesting prime-generating formula. It uses the con­stant 𝜉 and generates the primes in order!

The con­stant’s approximation.

$$\\x0000xi = 1.603502629914017832315523632362646507807932231768273436867961017532625344\dots$$$$\\x0000xi = 1.603502629914017832315523632362646507807932231768273436867961017532625344\dots$$

The formula $p_n$$p_n$ calculates the 𝑛-th prime.

$$p_n=\lfloor{10^{2\cdot n}\cdot\sqrt{\\x0000xi^3}}\rfloor-\lfloor{10^{2\cdot(n-1)}\cdot\sqrt{\\x0000xi^3}}\rfloor\cdot 10^2$$$$p_n=\lfloor{10^{2\cdot n}\cdot\sqrt{\\x0000xi^3}}\rfloor-\lfloor{10^{2\cdot(n-1)}\cdot\sqrt{\\x0000xi^3}}\rfloor\cdot 10^2$$

The first few values for $p_n$$p_n$ when starting with 𝑛 = 0 are as follows.

$$p_{0\text{ to }7}=\{2,3,5,7,11,13,17,19\}$$$$p_{0\text{ to }7}=\{2,3,5,7,11,13,17,19\}$$